Tagged Questions

Harmonic analysis is the generalisation of Fourier analysis. Use this tag for analysis on locally compact groups (e.g. Pontryagin duality), eigenvalues of the Laplacian on compact manifolds or graphs, and the abstract study of Fourier transform on Euclidean spaces (singular integrals, ...

Let $A$ be a Banach algebra and $I$ be an ideal of $A$. A derivation $D:A\to I$ is a linear bounded map, with the following property:
$$D(ab)=aD(b)+D(a)b,\qquad a,b\in A.$$
Suppose that $I$ is dense ...

I recently came across this inequality: Prove that for any $\zeta\in\mathbb{C}$, $\zeta\ne 0,1$, we have that
$$\int_0^1 \log|x-\zeta|dx\ge \frac{\log|\zeta|+\log|1-\zeta|}{2}-1.$$
How do you prove ...

Consider $H$ as a Hilbert space. How can I find a Banach space $E$, for that, $H=B(E)$ where $B(E)$ is the set of bounded linear operator on $E$?
(At least under some conditions on $H$)
Also if the ...

Let $f\in L^{\infty}(\mathbb{R}^{n})$ be supported in the unit ball and have mean zero. Let $\phi\in L^{1}(\mathbb{R}^{n})\cap C^{\alpha}(\mathbb{R}^{n})$ be a Holder continuous function with exponent ...

The Kelvin transform of the circle in $\mathbb{R}^n$ with centre $\textbf{u}$ and radius $r$ is defined by $$\textbf{y} \mapsto \textbf{u} + r^2|\textbf{y} - \textbf{u}|^{-2}(\textbf{y}-\textbf{u}).$$
...

Let $G$ be a locally compact group. Then we may snap our fingers, mention some measure theory results, and the group algebra $L^{1}(G)$ instantly appears.
Is there an example of the more generalized ...

It is known that if $f \in L^1(\mathbb R)$, $\widehat f(\xi) \neq 0$ for any $\xi \in \mathbb R$, then for any $h \in L^1(\mathbb R)$ such that $\widehat h$ is compactly supported there exists $g \in ...

I don't understand Lemma 2 of the paper Hilbert transforms along curves, II: A flat case by Michael Christ. The situation is as follows. I slightly simplified it from the exact context in the source.
...

Let $f: \mathbb R \rightarrow \mathbb R$ be a bounded function that is smooth and also in $L^1(\mathbb R) \cap L^2(\mathbb R)$. I want to prove that the Cauchy transform of this function $Kf$ is in ...